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Intertemporal Pricing Under Minimax Regret OPERATIONS RESEARCH Articles in Advance, pp. 1–25 ISSN 0030-364X (print) ISSN 1526-5463 (online) ó https://doi.org/10.1287/opre.2016.1548 © 2016 INFORMS Intertemporal Pricing Under Minimax Regret René Caldentey Booth School of Business, University of Chicago, Chicago, Illinois 60637, [email protected] Ying Liu, Ilan Lobel Stern School of Business, New York University, New York, New York 10012 {[email protected], [email protected]} We consider the pricing problem faced by a monopolist who sells a product to a population of consumers over a finite time horizon. Customers’ types differ along two dimensions: (i) their willingness-to-pay for the product and (ii) their arrival time during the selling season. We assume that the seller knows only the support of the customers’ valuations and do not make any other distributional assumptions about customers’ willingness-to-pay or arrival times. We consider a robust formulation of the seller’s pricing problem that is based on the minimization of her worst-case regret. We consider two distinct cases of customers’ purchasing behavior: myopic and strategic customers. For both of these cases, we characterize optimal price paths. For myopic customers, the regret is determined by the price at a critical time. Depending on the problem parameters, this critical time will be either the end of the selling season or it will be a time that equalizes the worst-case regret generated by undercharging customers and the worst-case regret generated by customers waiting for the price to fall. The optimal pricing strategy is not unique except at the critical time. For strategic consumers, we develop a robust mechanism design approach to compute an optimal policy. Depending on the problem parameters, the optimal policy might lead some consumers to wait until the end of the selling season and might price others out of the market. Under strategic customers, the optimal price equalizes the regrets generated by different customer types that arrive at the beginning of the selling season. We show that a seller that does not know if the customers are myopic should price as if they are strategic. We also show there is no benefit under myopic consumers to having a selling season longer than a certain uniform bound, but that the same is not true with strategic consumers. Keywords: demand uncertainty; strategic consumers; robust optimization; prior-free; worst-case regret. Subject classifications: marketing: pricing; games/group decisions: noncooperative; programming. Area of review: Operations and Supply Chains. History: Received November 2013; revisions received April 2015, April 2016; accepted June 2016. Published online in Articles in Advance November 7, 2016. 1. Introduction they typically do not have access to a data set that is rich Over the last couple of decades, dynamic pricing has been enough to estimate valuation and arrival time distributions. transformed from a curious and somewhat controversial The second reason is less obvious but equally important: taking the probability distributions of customer valuations practice used primarily by upstart airlines into a technique and arrival times as given assumes away a significant part that is widely used in a variety of industries. As technol- of the lack of knowledge about customer valuations and ogy has evolved and reduced menu costs, retailers of all arrival times. sorts have adopted intertemporal pricing practices. One of These issues are especially acute for firms introducing the key economic drivers behind the rapid dissemination new products into the marketplace. When firms launch new of dynamic pricing is demand uncertainty: there is enor- products, they usually have very little information about mous value for a firm in being able to change prices over how much customers are willing to pay for them. This great time in situations where the firm does not know how much degree of uncertainty makes new products excellent can- customers are willing to pay for its products. didates for dynamic pricing strategies. However, the same In response to the increasing use of dynamic pricing in uncertainty about customer valuations also hobbles our practice, academics have proposed a variety of techniques ability to use established dynamic pricing techniques since Downloaded from informs.org by [128.122.149.154] on 26 January 2017, at 18:23 . For personal use only, all rights reserved. for algorithmically determining intertemporal pricing poli- they rely on the firm knowing the probability distribution of cies. However, the vast majority of these approaches require customer valuations. Pricing of new products is no trivial the firm to know the probability distributions of customer matter. For instance, a McKinsey study reported that more valuations and arrival times. Assuming that the firm knows than a 100,000 new products are introduced yearly into the a full probabilistic model of customer valuations and arrival U.S. retail industry, but 70%–80% of these launches fail. times is problematic for at least two reasons. The first rea- The use of dynamic pricing for new products can reduce son is the obvious one: firms do not have access to such the impact of demand uncertainty on the firm and, in this probability distributions; even when they have sales data, way, help some of these launches succeed. 1 Caldentey, Liu, and Lobel: Intertemporal Pricing Under Minimax Regret 2 Operations Research, Articles in Advance, pp. 1–25, © 2016 INFORMS We approach the problem of intertemporal pricing from 1.1. Contributions the perspective of robust optimization (see Bertsimas and The primary contribution of this paper is the development Sim 2004, Ben-Tal et al. 2009). Specifically, we assume of a robust optimization methodology to compute intertem- that the seller only knows the range of customers’ valua- poral pricing policies that minimize a firm’s worst-case tion (or willingness to pay) for its product and makes no regret when selling to myopic or strategic consumers with additional assumptions about their distribution or about the uncertain valuations and arrival times. customers’ arrival process. The formulation we propose is In Section 4, we consider the case in which the firm sells quite parsimonious, but still sufficiently rich to give rise to to myopic customers. The regret from a given consumer different types of pricing policies for different sets of prob- can be decomposed into two terms: a valuation and a delay lem parameters. Our model formulation can be used both in regret. Valuation regret captures losses due to undercharg- settings with limited information, where the firm has only a ing consumers and can be lowered by raising prices. Delay very rough guess of the customer’s range of valuations, as regret captures losses due to consumers waiting for lower well as in an environment that is more data-rich, where the prices and can be reduced by lowering prices overall. For firm can use sales data to estimate an uncertainty set of the any given regret level R, there exists a price path that main- customers’ valuations. It is also flexible enough to allow tains the valuation regret for customers with high value at a us to study optimal pricing policies for myopic as well as constant R over the selling horizon and there exists another strategic customers, the latter being customers who time price path that maintains the delay regret for customers that their purchases to maximize their own discounted utilities. arrive at the beginning of the season at the same constant R. Under such a minimalist informational structure, the We show that if the time horizon is sufficiently long and the standard expected profit maximization criterion is not ap- market uncertainty is sufficiently high, the minimax regret propriate as the seller’s objective function (for more details, is determined by ensuring these two price paths intersect see Section 2). Instead, we consider the seller’s regret, tangentially for a given regret level R. The unique time which is defined as the difference between her payoff under where these two price paths intersect constitutes what we full demand information and her realized payoff. In this set- call the critical time. The optimal price offered at the crit- ting, an optimal pricing strategy is one that minimizes the ical time is uniquely determined, but generically there are difference between the seller’s ex-post payoff and that of a multiple optimal price paths. The price paths that maintain clairvoyant who sets prices knowing customer types (valu- constant valuation regret and delay regret determine the ation and arrival time) in advance. In particular, we assume boundaries of the set of optimal price paths. Any continu- the seller chooses a policy that minimizes her worst-case ous decreasing price path within these boundaries is opti- anticipated ex post regret. The seller assumes that nature mal, as long as the final price is below a certain value. A selects customer types from the uncertainty set to gener- typical optimal maximal price path includes an initial full- ate as much regret as possible, and that customers behave markup period where prices are set equal to the upper limit either myopically or strategically with respect to prices. of customers’ valuation range (v) followed by a markdown The first paper to propose a minimax regret criterion for period. In contrast, the optimal¯ minimal price path has no pricing without a prior distribution over customer valua- markup period and has less significant markdowns. In other tions was Bergemann and Schlag (2008). Our approach words, the seller has some flexibility to set prices either can be seen as an intertemporal version of Bergemann and aggressively (maximal solution) or conservatively (minimal Schlag (2008)’s static model. solution) during the early and late stages of the selling sea- We make several modeling assumptions that we wish to son, but not at an intermediate critical time.
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